scholarly journals ORDER AND ORDERING IN DISCRETE MATHEMATICS AND INFORMATICS

2021 ◽  
Vol 3 (1) ◽  
pp. 37-43
Author(s):  
V. K. Ovsyak ◽  
◽  
O. V. Ovsyak ◽  
J. V. Petruszka ◽  
◽  
...  
Keyword(s):  
Mathematical Logic ◽  
Graph Theory ◽  
Set Theory ◽  
Operator Algebra ◽  
Proof Theory ◽  
Cartesian Product ◽  
Object Oriented ◽  
Discrete Mathematics ◽  
Program Execution ◽  
System Of Algorithmic Algebras

The available means of ordering and sorting in some important sections of discrete mathematics and computer science are studied, namely: in the set theory, classical mathematical logic, proof theory, graph theory, POST method, system of algorithmic algebras, algorithmic languages of object-oriented and assembly programming. The Cartesian product of sets, ordered pairs and ordered n-s, the description by means of set theory of an ordered pair, which are performed by Wiener, Hausdorff and Kuratowski, are presented. The requirements as for the relations that order sets are described. The importance of ordering in classical mathematical logic and proof theory is illustrated by the examples of calculations of the truth values of logical formulas and formal derivation of a formula on the basis of inference rules and substitution rules. Ordering in graph theory is shown by the example of a block diagram of the Euclidean algorithm, designed to find the greatest common divisor of two natural numbers. The ordering and sorting of both the instructions formed by two, three and four ordered fields and the existing ordering of instructions in the program of Post method are described. It is shown that the program is formed by the numbered instructions with unique instruction numbers and the presence of the single instruction with number 1. The means of the system of algorithmic algebras, which are used to perform the ordering and sorting in the algorithm theory, are illustrated. The operations of the system of algorithmic algebras are presented, which include Boolean algebra operations generalized to the three-digit alphabet and operator operations of operator algebra. The properties of the composition operation are described, which is intended to describe the orderings of the operators of the operator algebra in the system of algorithmic algebras. The orderings executed by means of algorithmic programming languages are demonstrated by the hypothetical application of the modern object-oriented programming language C#. The program must contain only one method Main () from which the program execution begins. The ARM microprocessor assembly program must have only one ENTRY directive from which the program execution begins.

One hundred and two problems in mathematical logic

1975 ◽  
Vol 40 (2) ◽  
pp. 113-129 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
Logic Model ◽  
Mathematical Statement ◽  
Question Mark ◽  
Truth Value ◽  
Expository Paper

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

scholarly journals The Prospects for Mathematical Logic in the Twenty-First Century

2001 ◽  
Vol 7 (2) ◽  
pp. 169-196 ◽  
Author(s):  
Samuel R. Buss ◽  
Alexander S. Kechris ◽  
Anand Pillay ◽  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
First Century ◽  
Future Developments ◽  
The Future ◽  
Twenty First Century

AbstractThe four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

scholarly journals The use of discrete mathematics algorithms in technological design

2020 ◽  
Vol 329 ◽  
pp. 03082
Author(s):  
Nina Glinskaya ◽  
Isabella Belonovskaya
Keyword(s):  
Decision Making ◽  
Graph Theory ◽  
Structural Optimization ◽  
Set Theory ◽  
Traveling Salesman ◽  
Discrete Mathematics ◽  
Mechanical Processing ◽  
Mathematics Methods ◽  
Decision Making Processes ◽  
Technological Processing

The article examines the possibility of the formalization of the main stages of a detail mechanical processing using discrete mathematics methods. There are examples of using the graph theory for modeling practically all the stages of the development of processing technology. For structural optimization of technological process it is offered to use the problem of the traveling salesman. The choice of technological bases may be carried out both with using the graph theory and the set theory. The article makes a conclusion about the effectiveness of discrete mathematics methods for the formalization of decision making processes during technological processing.

Forcing in Proof Theory

2004 ◽  
Vol 10 (3) ◽  
pp. 305-333 ◽  
Author(s):  
Jeremy Avigad
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Model Theory ◽  
Proof Theory ◽  
Categorical Logic ◽  
Theory And Model ◽  
Number Of Branches

AbstractPaul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.

La théorie intuitionniste des types : sémantique des preuves et théorie des constructions

Dialogue ◽  
1997 ◽  
Vol 36 (2) ◽  
pp. 323-340
Author(s):  
Michel Bourdeau
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Type Theory ◽  
Proof Theory ◽  
Cartesian Product ◽  
Functional Space ◽  
Functional Dependency ◽  
Classical Approach ◽  
Constructive Theory ◽  
The Subject

AbstractMartin-Löf's constructive theory introduces, beside proof processes—the brouwerian mental construction—proof objects that could become the subject matter of a new kind of proof theory. In contradistinction to the classical approach, the proposition can then be defined as the set of its proofs. The lower level type theory is therefore a set theory, where the operators Σ and Π generalize the Cartesian product and the functional space to families of sets. To obtain the familiar logical constants, we have only to choose the logical reading of a : A. Σ and Π become ∃ and ∀, or, if there is no functional dependency, & and ⊃.

Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic

10.1142/12456 ◽  
2022 ◽  
Author(s):  
Douglas Cenzer ◽  
Jean Larson ◽  
Christopher Porter ◽  
Jindrich Zapletal
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Foundations Of Mathematics

A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

2021 ◽  
Vol 10 (3) ◽  
pp. 1-17
Author(s):  
Debabrata Mandal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Cartesian Product ◽  
Intuitionistic Fuzzy Set ◽  
Ideal Theory ◽  
Hesitant Fuzzy Set ◽  
Neutrosophic Set ◽  
Intuitionistic Fuzzy ◽  
Interval Valued ◽  
The Ideal

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.

Finding Dead-Point Positions of Planar Pin-Connected Linkages Through Graph Theoretical Duality Principle

2005 ◽  
Vol 128 (3) ◽  
pp. 599-609 ◽  
Author(s):  
O. Shai ◽  
I. Polansky
Keyword(s):  
Graph Theory ◽  
Discrete Mathematics ◽  
Duality Principle ◽  
Mathematical Foundation ◽  
The Dead

The paper brings another view on detecting the dead-point positions of an arbitrary planar pin-connected linkage by employing the duality principle of graph theory. It is first shown how the dead-point positions are derived through the interplay between the linkage and its dual determinate truss—the relation developed in the previous works by means of graph theory. At the next stage, the process is shown to be performed solely upon the linkage by employing a new variable, the dual of potential, termed face force. Since the mathematical foundation of the presented method is discrete mathematics, the paper points to possible computerization of the method.

scholarly journals VIRASORO CORRELATION FUNCTIONS FOR VERTEX OPERATOR ALGEBRAS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250106 ◽  
Author(s):  
DONNY HURLEY ◽  
MICHAEL P. TUITE
Keyword(s):  
Graph Theory ◽  
Vertex Operator ◽  
Generating Functions ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Genus Zero ◽  
Genus One

We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.

ASSOCIATION-BASED INFORMATION FLOW CONTROL IN OBJECT-ORIENTED SYSTEMS

2004 ◽  
Vol 14 (03) ◽  
pp. 291-322
Author(s):  
SHIH-CHIEN CHOU ◽  
YING-KAI WEN
Keyword(s):  
Flow Control ◽  
Information Flow ◽  
Information Leakage ◽  
Object Oriented ◽  
Control Model ◽  
Program Execution ◽  
Information Flows ◽  
Information Flow Control ◽  
Object Oriented Systems

Controlling information flows to prevent information leakage within an application is essential. According to the maturity of object-oriented techniques, many models were developed for the control in object-oriented systems. Since objects may be dynamically instantiated during program execution, controlling information flows among objects is difficult. Our research revealed that association is useful in the control. We developed an association-based information flow control model for object-oriented systems. It precisely controls information flows among objects through associations and constraints. It also offers features such as controlling method invocation through argument sensitivity, allowing declassification, allowing purpose-oriented method invocation, and precisely controlling write access. This paper proposes the model and the implementation of the model, which is composed of the language AbFlow (association-based flow) and its supporting environment.

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